A linear programming computer package is needed. Western Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the steakhouse operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4-hour shifts during peak meal times. On Saturdays the steakhouse is open from 11:00 a.m. to 10:00 p.m. Management wants to develop a schedule for part-time employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $7.60 per hour. The total number of full-time and part-time employees needed varies with the time of day as shown. Time Number of Employees Needed 11:00 a.m.–Noon 9 Noon–1:00 p.m. 9 1:00 p.m.–2:00 p.m. 9 2:00 p.m.–3:00 p.m. 3 3:00 p.m.–4:00 p.m. 3 4:00 p.m.–5:00 p.m. 3 5:00 p.m.–6:00 p.m. 6 6:00 p.m.–7:00 p.m. 12 7:00 p.m.–8:00 p.m. 12 8:00 p.m.–9:00 p.m. 7 9:00 p.m.–10:00 p.m. 7 One full-time employee comes on duty at 11:00 a.m., works 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00 p.m. and works the same 4-hours-on, 1-hour-off, 4-hours-on pattern. (a) Develop a minimum-cost schedule for part-time employees. (Let  xi  = the number of part-time employees who start work beginning at hour i where  i = 1 = 11:00 a.m.,   i = 2 = Noon,  etc). Min        s.t.11:00 a.m.       Noon       1:00 p.m.       2:00 p.m.       3:00 p.m.       4:00 p.m.       5:00 p.m.       6:00 p.m.       7:00 p.m.       8:00 p.m.       9:00 p.m.       x1, x2, x3, x4, x5, x6, x7, x8 ≥ 0 Find the optimal solution. (x1, x2, x3, x4, x5, x6, x7, x8) =                (b) What is the total payroll (in $) for the part-time employees? $  How many part-time shifts are needed?  shifts Use the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts. This answer has not been graded yet. (c) Assume that part-time employees can be assigned either a 3-hour or a 4-hour shift. Develop a minimum-cost schedule for the part-time employees. (Let  xi  be the same variables from part a. Let  yi  = the number of part-time employees who start work beginning at hour i where  i = 1 = 11:00 a.m.,   i = 2 = Noon,  etc). Min        s.t.11:00 a.m.       Noon       1:00 p.m.       2:00 p.m.       3:00 p.m.       4:00 p.m.       5:00 p.m.       6:00 p.m.       7:00 p.m.       8:00 p.m.       9:00 p.m.       x1, x2, x3, x4, x5, x6, x7, x8, y1, y2, y3, y4, y5, y6, y7, y8, y9 ≥ 0 Find the optimal solution. (x1, x2, x3, x4, x5, x6, x7, x8, y1, y2, y3, y4, y5, y6, y7, y8, y9) =                How many part-time shifts are needed, and what is the cost savings (in $) compared to the previous schedule? part-time shifts neededcost savings$

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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A linear programming computer package is needed.
Western Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the steakhouse operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4-hour shifts during peak meal times. On Saturdays the steakhouse is open from 11:00 a.m. to 10:00 p.m. Management wants to develop a schedule for part-time employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $7.60 per hour. The total number of full-time and part-time employees needed varies with the time of day as shown.
Time Number of Employees Needed
11:00 a.m.–Noon 9
Noon–1:00 p.m. 9
1:00 p.m.–2:00 p.m. 9
2:00 p.m.–3:00 p.m. 3
3:00 p.m.–4:00 p.m. 3
4:00 p.m.–5:00 p.m. 3
5:00 p.m.–6:00 p.m. 6
6:00 p.m.–7:00 p.m. 12
7:00 p.m.–8:00 p.m. 12
8:00 p.m.–9:00 p.m. 7
9:00 p.m.–10:00 p.m. 7
One full-time employee comes on duty at 11:00 a.m., works 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00 p.m. and works the same 4-hours-on, 1-hour-off, 4-hours-on pattern.
(a)
Develop a minimum-cost schedule for part-time employees. (Let 
xi
 = the number of part-time employees who start work beginning at hour i where 
i = 1 = 11:00 a.m.,
 
i = 2 = Noon,
 etc).
Min 
 
 
 
s.t.11:00 a.m.
 
 
 
Noon
 
 
 
1:00 p.m.
 
 
 
2:00 p.m.
 
 
 
3:00 p.m.
 
 
 
4:00 p.m.
 
 
 
5:00 p.m.
 
 
 
6:00 p.m.
 
 
 
7:00 p.m.
 
 
 
8:00 p.m.
 
 
 
9:00 p.m.
 
 
 
x1, x2, x3, x4, x5, x6, x7, x8 ≥ 0
Find the optimal solution.
(x1, x2, x3, x4, x5, x6, x7, x8) = 
 
 
 
 
 
 
 
 
(b)
What is the total payroll (in $) for the part-time employees?
How many part-time shifts are needed?
 shifts
Use the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts.

This answer has not been graded yet.

(c)
Assume that part-time employees can be assigned either a 3-hour or a 4-hour shift. Develop a minimum-cost schedule for the part-time employees. (Let 
xi
 be the same variables from part a. Let 
yi
 = the number of part-time employees who start work beginning at hour i where 
i = 1 = 11:00 a.m.,
 
i = 2 = Noon,
 etc).
Min 
 
 
 
s.t.11:00 a.m.
 
 
 
Noon
 
 
 
1:00 p.m.
 
 
 
2:00 p.m.
 
 
 
3:00 p.m.
 
 
 
4:00 p.m.
 
 
 
5:00 p.m.
 
 
 
6:00 p.m.
 
 
 
7:00 p.m.
 
 
 
8:00 p.m.
 
 
 
9:00 p.m.
 
 
 
x1, x2, x3, x4, x5, x6, x7, x8, y1, y2, y3, y4, y5, y6, y7, y8, y9 ≥ 0
Find the optimal solution.
(x1, x2, x3, x4, x5, x6, x7, x8, y1, y2, y3, y4, y5, y6, y7, y8, y9) = 
 
 
 
 
 
 
 
 
How many part-time shifts are needed, and what is the cost savings (in $) compared to the previous schedule?
part-time shifts neededcost savings$
A linear programming computer package is needed.
Western Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the steakhouse operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4-hour shifts during peak meal times. On Saturdays the steakhouse is open from 11:00 a.m. to 10:00 p.m. Management wants to develop a schedule for part-time employees that
will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $7.60 per hour. The total number of full-time and part-time employees needed varies with the time of day as shown.
Min
s.t.
11:00 a.m.-Noon
Noon-1:00 p.m.
1:00 p.m.-2:00 p.m.
2:00 p.m.-3:00 p.m.
3:00 p.m.-4:00 p.m.
4:00 p.m.-5:00p.m.
5:00 p.m.-6:00 p.m.
6:00 p.m.-7:00 p.m.
7:00 p.m.-8:00 p.m.
8:00 p.m.-9:00 p.m.
9:00 p.m.-10:00 p.m.
11:00 a.m.
Noon
1:00p.m.
2:00 p.m.
3:00 p.m.
4:00 p.m.
Time
5:00 p.m.
6:00 p.m.
7:00 p.m.
8:00 p.m.
One full-time employee comes on duty at 11:00 a.m., works 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00 p.m. and works the same 4-hours-on, 1-hour-off, 4-hours-on pattern.
(a) Develop a minimum-cost schedule for part-time employees. (Let x, = the number of part-time employees who start work beginning at hour / where / = 1 = 11:00 a.m.; i = 2 = Noon, etc).
Number of
Employees
Needed
9:00 p.m.
X₁ X₂ X3 X4 X₁ X 6 X y XB 2.0
9
9
9
3
3
3
6
12
12
7
7
Transcribed Image Text:A linear programming computer package is needed. Western Family Steakhouse offers a variety of low-cost meals and quick service. Other than management, the steakhouse operates with two full-time employees who work 8 hours per day. The rest of the employees are part-time employees who are scheduled for 4-hour shifts during peak meal times. On Saturdays the steakhouse is open from 11:00 a.m. to 10:00 p.m. Management wants to develop a schedule for part-time employees that will minimize labor costs and still provide excellent customer service. The average wage rate for the part-time employees is $7.60 per hour. The total number of full-time and part-time employees needed varies with the time of day as shown. Min s.t. 11:00 a.m.-Noon Noon-1:00 p.m. 1:00 p.m.-2:00 p.m. 2:00 p.m.-3:00 p.m. 3:00 p.m.-4:00 p.m. 4:00 p.m.-5:00p.m. 5:00 p.m.-6:00 p.m. 6:00 p.m.-7:00 p.m. 7:00 p.m.-8:00 p.m. 8:00 p.m.-9:00 p.m. 9:00 p.m.-10:00 p.m. 11:00 a.m. Noon 1:00p.m. 2:00 p.m. 3:00 p.m. 4:00 p.m. Time 5:00 p.m. 6:00 p.m. 7:00 p.m. 8:00 p.m. One full-time employee comes on duty at 11:00 a.m., works 4 hours, takes an hour off, and returns for another 4 hours. The other full-time employee comes to work at 1:00 p.m. and works the same 4-hours-on, 1-hour-off, 4-hours-on pattern. (a) Develop a minimum-cost schedule for part-time employees. (Let x, = the number of part-time employees who start work beginning at hour / where / = 1 = 11:00 a.m.; i = 2 = Noon, etc). Number of Employees Needed 9:00 p.m. X₁ X₂ X3 X4 X₁ X 6 X y XB 2.0 9 9 9 3 3 3 6 12 12 7 7
8:00 p.m.
9:00 p.m.
Find the optimal solution.
(Ху Жус Хз. Жді Х.s, X,
(b) What is the total payroll (in s) for the part-time employees?
How many part-time shifts are needed?
shifts
Use the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts.
This answer has not been graded yet.
(c) Assume that part-time employees can be assigned either a 3-hour or a 4-hour shift. Develop a minimum-cost schedule for the part-time employees. (Let x, be the same variables from part a. Let y, the number of part-time employees who start work beginning at hour / where i = 1 = 11:00 a.m., / = 2 = Noon, etc).
Mini
s.t.
11:00 a.m.
Noon
1:00 p.m.
2:00 p.m.
3:00 p.m.
4:00 p.m.
x₂) =
5:00 p.m.
6:00 p.m.
7:00 p.m.
8:00 p.m.
9:00 p.m.
X₁ X₂ X3 X4 X X X X Y ₁ Y ₂ Y3 Yar Y5 Y6 Y7 Yg Yg 20
Find the optimal solution.
(X1 X2 X3 X4 X5 X
5 Y6 Y7 YB Yg) =
How many part-time shifts are needed, and what is the cost savings (in $) compared to the previous schedule?
part-time shifts needed
cost savings
Transcribed Image Text:8:00 p.m. 9:00 p.m. Find the optimal solution. (Ху Жус Хз. Жді Х.s, X, (b) What is the total payroll (in s) for the part-time employees? How many part-time shifts are needed? shifts Use the surplus variables to comment on the desirability of scheduling at least some of the part-time employees for 3-hour shifts. This answer has not been graded yet. (c) Assume that part-time employees can be assigned either a 3-hour or a 4-hour shift. Develop a minimum-cost schedule for the part-time employees. (Let x, be the same variables from part a. Let y, the number of part-time employees who start work beginning at hour / where i = 1 = 11:00 a.m., / = 2 = Noon, etc). Mini s.t. 11:00 a.m. Noon 1:00 p.m. 2:00 p.m. 3:00 p.m. 4:00 p.m. x₂) = 5:00 p.m. 6:00 p.m. 7:00 p.m. 8:00 p.m. 9:00 p.m. X₁ X₂ X3 X4 X X X X Y ₁ Y ₂ Y3 Yar Y5 Y6 Y7 Yg Yg 20 Find the optimal solution. (X1 X2 X3 X4 X5 X 5 Y6 Y7 YB Yg) = How many part-time shifts are needed, and what is the cost savings (in $) compared to the previous schedule? part-time shifts needed cost savings
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